Definite Integral from 0 to Half Pi of Odd Power of Cosine x/Proof 2
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Theorem
- $\ds \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$
Proof
\(\ds \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x\) | \(=\) | \(\ds \int_0^{\frac \pi 2} \paren {\sin x}^{\frac 2 2 - 1} \paren {\cos x}^{2 \paren {n + 1} - 1} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \map \Beta {\frac 1 2, n + 1}\) | Definition 2 of Beta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \cdot \frac {\map \Gamma {n + 1} \map \Gamma {\frac 1 2} } {\map \Gamma {n + 1 + \frac 1 2} }\) | Definition 3 of Beta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \cdot \frac {n! \sqrt \pi} {\map \Gamma {n + 1 + \frac 1 2} }\) | Gamma Function Extends Factorial, Gamma Function of One Half | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \cdot n! \sqrt \pi \cdot \frac{2^{2 n + 2} \paren {n + 1}!} {\paren {2 n + 2}! \sqrt \pi}\) | Gamma Function of Positive Half-Integer | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n! \cdot 2^{2 n + 1} \paren {n + 1}!} {\paren {2 n + 2} \paren {2 n + 1}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 2 \cdot \frac {n! \cdot 2^{2 n} \paren {n + 1} n!} {\paren {n + 1} \paren {2 n + 1}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {2^n n!}^2} {\paren {2 n + 1}!}\) |
$\blacksquare$